Conchoid
The conchoid (the "shell-like" - like Latin concha from Greek κόγχη or κόγχος, shell ) is a special flat curve. It describes the movement of a point which - viewed from a fixed point ( pole ) - maintains a constant distance from a given curve.
Actual conchoid
It was already known in ancient Greece and is called the conchoid of Nicomedes after Nicomedes . Another name is Shell Curve . The name is derived from the fact that the graph resembles the two shells of a clam.
properties
- The points of the conchoid of Nicomedes are characterized by the following geometric property: Given are a straight line g ("leading straight line"), a point A that is at a distance of a from g (with a> 0), and a real number b (with b> 0). Then for an arbitrary point B of the straight line g the two points P and P ', which lie on the straight line AB and are at the distance b from B, lie on the conchoid.
- The falls are a type of curve that is known by the common name dog curve , especially for one branch that resembles the actual tractrix .
In the following it is assumed that the coordinate axes are as in the sketch, i.e. the pole is at the origin .
- The conchoid of the Nicomedes is axially symmetrical with respect to the x-axis. In general, three points of the curve lie on the axis of symmetry, namely , and the origin.
- For the origin is an isolated point .
- For two of the three points at the origin coincide, for the origin is a colon of the curve, so it is run through twice, the graph has a loop.
- The two tangents at the origin have the equations
-
and .
- For both tangents coincide with the x-axis. So the origin is a real tip
Ordinary conchoid
The term conchoid can be generalized:
A curve k ( leading curve ), a point A ( pole ) and a positive real number b are given. For any point B that lies on curve k, consider the two points that lie on straight line AB and are at distance b from B. The set of all these points is called the conchoid of the guide curve .
The simplest representation uses polar coordinates: If A is at the origin and if , then the equation of the ordinary conchoid is:
properties
All common conchoids are cissoids , with one curve being a circle at the origin.
A Pascal snail is a conchoid, where the given curve is a circle.
General conchoid
If one extends the formation rule by not plotting the distance b along the straight line AB, but along a straight line that has a constant angle to AB at point B , one obtains the general conchoid . In the case of and the usual conchoid results, otherwise one speaks of a crooked conchoid .
Conchoidal toothing in gear technology
In the transmission art is the so called Konchoidenverzahnung one of several techniques for teeth of gears and toothed racks .
Remarks
- De Sluze's so-called conchoid is actually a special cissoid .
- Dürer's so-called conchoid is a more general construction.
Web links
- Johanneum Lüneburg dog curve or conchoid of Nicomedes ( Memento from July 3, 2007 in the Internet Archive )